As such, algebraic manifolds are a generalisation of the concept of smooth curves and surfaces defined by polynomials.
An example is the sphere, which can be defined as the zero set of the polynomial x2 + y2 + z2 – 1, and hence is an algebraic variety.
For an algebraic manifold, the ground field will be the real numbers or complex numbers; in the case of the real numbers, the manifold of real points is sometimes called a Nash manifold.
Every sufficiently small local patch of an algebraic manifold is isomorphic to km where k is the ground field.
Equivalently the variety is smooth (free from singular points).